= Solution
For radial <null geodesics>, the radial metric is $-fdt^2+dr^2/f$. Define the <tortoise coordinate> by
$$
\boxed{r_* =\int^r\frac{\sqrt{H(\rho)}}{F(\rho)}\,d\rho
=\int^r\frac{\sqrt{\rho(\rho+q)}}{\rho-2m}\,d\rho.}
$$
Then the radial metric becomes $f(-dt^2+dr_*^2)=-f\,du\,dv$. Its two null directions have $dt=\pm dr_*$, so $\boxed{u=t-r_*\text{ is constant on outgoing rays},\quad v=t+r_*\text{ on ingoing rays}}$ in the exterior. The radial null curves are null geodesics up to reparametrization: in the two-dimensional radial metric a null tangent's acceleration is proportional to itself, and spherical symmetry introduces no angular acceleration.
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